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Rational points on fibrations with few non-split fibres

  • Yonatan Harpaz , Dasheng Wei and Olivier Wittenberg EMAIL logo
Published/Copyright: August 12, 2022

Abstract

We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is 2 , as well as in various instances where it is 3. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel’s hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer–Manin obstruction in families.

Award Identifier / Grant number: 11622111

Award Identifier / Grant number: 11631009

Funding statement: The second-named author is supported by National Key R&D Program of China and National Natural Science Foundation of China (Grant Nos. 11622111 and 11631009).

Acknowledgements

We thank the anonymous referee for their helpful comments.

References

[1] M. Artin, Faisceaux constructibles, cohomologie d’une courbe algébrique, Exp. IX, Théorie des topos et cohomologie étale des schémas. Séminaire de géométrie algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Math. 305, Springer, Berlin (1973), 1–42. 10.1007/BFb0070715Search in Google Scholar

[2] H.-J. Bartels, Zur Arithmetik von Diedergruppenerweiterungen, Math. Ann. 256 (1981), no. 4, 465–473. 10.1007/BF01450542Search in Google Scholar

[3] M. Borovoi, C. Demarche and D. Harari, Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 4, 651–692. 10.24033/asens.2198Search in Google Scholar

[4] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin 1990. 10.1007/978-3-642-51438-8Search in Google Scholar

[5] K. S. Brown, Cohomology of groups, Grad. Texts in Math. 87, Springer, New York 1994. Search in Google Scholar

[6] T. D. Browning and D. R. Heath-Brown, Quadratic polynomials represented by norm forms, Geom. Funct. Anal. 22 (2012), no. 5, 1124–1190. 10.1007/s00039-012-0168-5Search in Google Scholar

[7] T. D. Browning and L. Matthiesen, Norm forms for arbitrary number fields as products of linear polynomials, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 6, 1383–1446. 10.24033/asens.2348Search in Google Scholar

[8] T. D. Browning, L. Matthiesen and A. N. Skorobogatov, Rational points on pencils of conics and quadrics with many degenerate fibers, Ann. of Math. (2) 180 (2014), no. 1, 381–402. 10.4007/annals.2014.180.1.8Search in Google Scholar

[9] T. D. Browning and D. Schindler, Strong approximation and a conjecture of Harpaz and Wittenberg, Int. Math. Res. Not. IMRN 2019 (2019), no. 14, 4340–4369. 10.1093/imrn/rnx252Search in Google Scholar

[10] Y. Cao, D. Wei and F. Xu, Strong approximation for a family of norm varieties, preprint (2018), https://arxiv.org/abs/1803.11003. 10.29007/584lSearch in Google Scholar

[11] Y. Cao and F. Xu, Strong approximation with Brauer–Manin obstruction for toric varieties, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 1879–1908. 10.5802/aif.3199Search in Google Scholar

[12] J.-L. Colliot-Thélène, Points rationnels sur les fibrations, Higher dimensional varieties and rational points (Budapest 2001), Bolyai Soc. Math. Stud. 12, Springer, Berlin (2003), 171–221. 10.1007/978-3-662-05123-8_7Search in Google Scholar

[13] J.-L. Colliot-Thélène, Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences, Arithmetic geometry, Lecture Notes in Math. 2009, Springer, Berlin (2011), 1–44. 10.1007/978-3-642-15945-9_1Search in Google Scholar

[14] J.-L. Colliot-Thélène, Un calcul de groupe de Brauer et une application arithmétique, Arithmetic and geometry, London Math. Soc. Lecture Note Ser. 420, Cambridge University, Cambridge (2015), 188–192. 10.1017/CBO9781316106877.012Search in Google Scholar

[15] J.-L. Colliot-Thélène and D. Harari, Approximation forte en famille, J. reine angew. Math. 710 (2016), 173–198. 10.1515/crelle-2013-0092Search in Google Scholar

[16] J.-L. Colliot-Thélène, D. Harari and A. N. Skorobogatov, Valeurs d’un polynôme à une variable représentées par une norme, Number theory and algebraic geometry, London Math. Soc. Lecture Note Ser. 303, Cambridge University, Cambridge (2003), 69–89. 10.1017/CBO9780511734946.006Search in Google Scholar

[17] J.-L. Colliot-Thélène and J.-J. Sansuc, Sur le principe de Hasse et l’approximation faible, et sur une hypothèse de Schinzel, Acta Arith. 41 (1982), no. 1, 33–53. 10.4064/aa-41-1-33-53Search in Google Scholar

[18] J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492. 10.1215/S0012-7094-87-05420-2Search in Google Scholar

[19] J.-L. Colliot-Thélène and J.-J. Sansuc, Principal homogeneous spaces under flasque tori: Applications, J. Algebra 106 (1987), no. 1, 148–205. 10.1016/0021-8693(87)90026-3Search in Google Scholar

[20] J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces. I, J. reine angew. Math. 373 (1987), 37–107. 10.1515/crll.1987.373.37Search in Google Scholar

[21] J.-L. Colliot-Thélène and A. N. Skorobogatov, Descent on fibrations over 𝐏 k 1 revisited, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 3, 383–393. 10.1017/S0305004199004077Search in Google Scholar

[22] J.-L. Colliot-Thélène and A. N. Skorobogatov, Good reduction of the Brauer–Manin obstruction, Trans. Amer. Math. Soc. 365 (2013), no. 2,579–590. 10.1090/S0002-9947-2012-05556-5Search in Google Scholar

[23] J.-L. Colliot-Thélène and A. N. Skorobogatov, The Brauer–Grothendieck group, Ergeb. Math. Grenzgeb. (3) 71, Springer, Cham 2021. 10.1007/978-3-030-74248-5Search in Google Scholar

[24] J.-L. Colliot-Thélène, A. N. Skorobogatov and P. Swinnerton-Dyer, Rational points and zero-cycles on fibred varieties: Schinzel’s hypothesis and Salberger’s device, J. reine angew. Math. 495 (1998), 1–28. 10.1515/crll.1998.019Search in Google Scholar

[25] J.-L. Colliot-Thélène and P. Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi–Brauer and similar varieties, J. reine angew. Math. 453 (1994), 49–112. 10.1515/crll.1994.453.49Search in Google Scholar

[26] B. Conrad, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257; erratum, J. Ramanujan Math. Soc. 24 (2009), no. 4, 427–428. Search in Google Scholar

[27] O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer, New York 2001. 10.1007/978-1-4757-5406-3Search in Google Scholar

[28] P. Deligne, Théorèmes de finitude en cohomologie -adique, Cohomologie étale, Lecture Notes in Math. 569, Springer, Berlin (1977), 233–261. 10.1007/BFb0091524Search in Google Scholar

[29] P. Deligne, Le théorème de plongement de Nagata, Kyoto J. Math. 50 (2010), no. 4, 661–670. 10.1215/0023608X-2010-009Search in Google Scholar

[30] U. Derenthal, A. Smeets and D. Wei, Universal torsors and values of quadratic polynomials represented by norms, Math. Ann. 361 (2015), no. 3–4, 1021–1042. 10.1007/s00208-014-1106-7Search in Google Scholar

[31] Y. A. Drakokhrust and V. P. Platonov, The Hasse norm principle for algebraic number fields, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 946–968. 10.1070/IM1987v029n02ABEH000971Search in Google Scholar

[32] T. Ekedahl, An effective version of Hilbert’s irreducibility theorem, Séminaire de théorie des nombres (Paris 1988–1989), Progr. Math. 91, Birkhäuser, Boston (1990), 241–249. Search in Google Scholar

[33] T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. 10.1090/S0894-0347-02-00402-2Search in Google Scholar

[34] G. Gras, Class field theory, Springer Monogr. Math., Springer, Berlin 2003. 10.1007/978-3-662-11323-3Search in Google Scholar

[35] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. 10.4007/annals.2008.167.481Search in Google Scholar

[36] B. Green and T. Tao, Linear equations in primes, Ann. of Math. (2) 171 (2010), no. 3, 1753–1850. 10.4007/annals.2010.171.1753Search in Google Scholar

[37] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), no. 2, 541–566. 10.4007/annals.2012.175.2.3Search in Google Scholar

[38] B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers U s + 1 [ N ] -norm, Ann. of Math. (2) 176 (2012), no. 2, 1231–1372. 10.4007/annals.2012.176.2.11Search in Google Scholar

[39] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 20 (1964), 5–259. 10.1007/BF02684747Search in Google Scholar

[40] A. Grothendieck, Le groupe de Brauer. III. Exemples et compléments, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam (1968), 88–188.Search in Google Scholar

[41] D. Handel, On products in the cohomology of the dihedral groups, Tohoku Math. J. (2) 45 (1993), no. 1, 13–42. 10.2748/tmj/1178225952Search in Google Scholar

[42] D. Harari, Méthode des fibrations et obstruction de Manin, Duke Math. J. 75 (1994), no. 1, 221–260. 10.1215/S0012-7094-94-07507-8Search in Google Scholar

[43] D. Harari, Flèches de spécialisations en cohomologie étale et applications arithmétiques, Bull. Soc. Math. France 125 (1997), no. 2, 143–166. 10.24033/bsmf.2302Search in Google Scholar

[44] D. Harari, Galois cohomology and class field theory, Universitext, Springer, Cham 2020. 10.1007/978-3-030-43901-9Search in Google Scholar

[45] D. Harari and A. N. Skorobogatov, Descent theory for open varieties, Torsors, étale homotopy and applications to rational points, London Math. Soc. Lecture Note Ser. 405, Cambridge University, Cambridge (2013), 250–279. 10.1017/CBO9781139525350.009Search in Google Scholar

[46] Y. Harpaz, A. N. Skorobogatov and O. Wittenberg, The Hardy–Littlewood conjecture and rational points, Compos. Math. 150 (2014), no. 12, 2095–2111. 10.1112/S0010437X14007568Search in Google Scholar

[47] Y. Harpaz and O. Wittenberg, On the fibration method for zero-cycles and rational points, Ann. of Math. (2) 183 (2016), no. 1, 229–295. 10.4007/annals.2016.183.1.5Search in Google Scholar

[48] D. R. Heath-Brown and B. Z. Moroz, On the representation of primes by cubic polynomials in two variables, Proc. Lond. Math. Soc. (3) 88 (2004), no. 2, 289–312. 10.1112/S0024611503014497Search in Google Scholar

[49] R. Heath-Brown and A. Skorobogatov, Rational solutions of certain equations involving norms, Acta Math. 189 (2002), no. 2, 161–177. 10.1007/BF02392841Search in Google Scholar

[50] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. 79 (1964), 205–326. 10.2307/1970547Search in Google Scholar

[51] A. J. Irving, Cubic polynomials represented by norm forms, J. reine angew. Math. 723 (2017), 217–250. 10.1515/crelle-2014-0070Search in Google Scholar

[52] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin 1996. 10.1007/978-3-662-03276-3Search in Google Scholar

[53] B. E. Kunyavskiĭ, Arithmetic properties of three-dimensional algebraic tori, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 102–107; translation in J. Sov. Math. 26 (1984), 1898–1901. 10.1007/BF01670577Search in Google Scholar

[54] S. Lang, Fundamentals of Diophantine geometry, Springer, New York 1983. 10.1007/978-1-4757-1810-2Search in Google Scholar

[55] S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. 10.2307/2372655Search in Google Scholar

[56] A. Macedo, The Hasse norm principle for A n -extensions, J. Number Theory 211 (2020), 500–512. 10.1016/j.jnt.2019.10.020Search in Google Scholar

[57] Y. I. Manin, Le groupe de Brauer–Grothendieck en géométrie diophantienne, Actes du congrès international des mathématiciens. Tome 1 (Nice 1970), Gauthier-Villars, Paris (1971), 401–411. 10.1142/9789812830517_0009Search in Google Scholar

[58] L. Matthiesen, On the square-free representation function of a norm form and nilsequences, J. Inst. Math. Jussieu 17 (2018), no. 1, 107–135. 10.1017/S1474748015000389Search in Google Scholar

[59] J. S. Milne, Étale cohomology, Princeton Math. Ser. 33, Princeton University, Princeton 1980. Search in Google Scholar

[60] M. Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 1–10. 10.1215/kjm/1250524969Search in Google Scholar

[61] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields, 2nd ed., Grundlehren Math. Wiss. 323, Springer, Berlin 2008. 10.1007/978-3-540-37889-1Search in Google Scholar

[62] L. B. Nisnevič, On the number of points of an algebraic manifold in a prime finite field, Dokl. Akad. Nauk SSSR (N. S.) 99 (1954), 17–20. Search in Google Scholar

[63] M. Raynaud, Revêtements étales et groupe fondamental. Séminaire de géométrie algébrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Math. 224, Société Mathématique de France, Paris 1971. Search in Google Scholar

[64] M. Rosenlicht, Some rationality questions on algebraic groups, Ann. Mat. Pura Appl. (4) 43 (1957), 25–50. 10.1007/BF02411903Search in Google Scholar

[65] J.-P. Serre, Résumé des cours au Collège de France, 1991–1992. Search in Google Scholar

[66] A. Shute, Polynomials represented by norm forms via the beta sieve, preprint (2022). Search in Google Scholar

[67] A. N. Skorobogatov, On the fibration method for proving the Hasse principle and weak approximation, Séminaire de théorie des nombres (Paris 1988–1989), Progr. Math. 91, Birkhäuser, Boston (1990), 205–219. Search in Google Scholar

[68] A. N. Skorobogatov, Descent on fibrations over the projective line, Amer. J. Math. 118 (1996), no. 5, 905–923. 10.1353/ajm.1996.0045Search in Google Scholar

[69] A. N. Skorobogatov, Torsors and rational points, Cambridge Tracts in Math. 144, Cambridge University, Cambridge 2001. 10.1017/CBO9780511549588Search in Google Scholar

[70] A. N. Skorobogatov, Descent on toric fibrations, Arithmetic and geometry, London Math. Soc. Lecture Note Ser. 420, Cambridge University, Cambridge (2015), 422–435. 10.1017/CBO9781316106877.022Search in Google Scholar

[71] A. Smeets, Principes locaux-globaux pour certaines fibrations en torseurs sous un tore, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 1, 131–145. 10.1017/S0305004114000577Search in Google Scholar

[72] J. Stix, Trading degree for dimension in the section conjecture: The non-abelian Shapiro lemma, Math. J. Okayama Univ. 52 (2010), 29–43. Search in Google Scholar

[73] M. Swarbrick Jones, A note on a theorem of Heath–Brown and Skorobogatov, Q. J. Math. 64 (2013), no. 4, 1239–1251. 10.1093/qmath/has031Search in Google Scholar

[74] P. Swinnerton-Dyer, Rational points on pencils of conics and on pencils of quadrics, J. Lond. Math. Soc. (2) 50 (1994), no. 2, 231–242. 10.1112/jlms/50.2.231Search in Google Scholar

[75] P. Swinnerton-Dyer, Rational points on some pencils of conics with 6 singular fibres, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), no. 2, 331–341. 10.5802/afst.935Search in Google Scholar

[76] A. Várilly-Alvarado and B. Viray, Higher-dimensional analogs of Châtelet surfaces, Bull. Lond. Math. Soc. 44 (2012), no. 1, 125–135; erratum, Bull. Lond. Math. Soc. 47 (2015), no. 2, 217–218. 10.1112/blms/bdr075Search in Google Scholar

[77] V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Transl. Math. Monogr. 179, American Mathematical Society, Providence 1998. Search in Google Scholar

[78] V. E. Voskresenskiĭ and B. E. Kunyavskiĭ, Maximal tori in semisimple algebraic groups, preprint (1984), VINITI, no. 1269-84. Search in Google Scholar

[79] D. Wei, On the equation N K / k ( Ξ ) = P ( t ) , Proc. Lond. Math. Soc. (3) 109 (2014), no. 6, 1402–1434. 10.1112/plms/pdu035Search in Google Scholar

[80] D. Wei, The unramified Brauer group of norm one tori, Adv. Math. 254 (2014), 642–663. 10.1016/j.aim.2013.12.016Search in Google Scholar

[81] D. S. Wei, Strong approximation for a toric variety, Acta Math. Sin. (Engl. Ser.) 37 (2021), no. 1, 95–103. 10.1007/s10114-021-8193-7Search in Google Scholar

[82] O. Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields, Algebraic geometry (Salt Lake City 2015), Proc. Sympos. Pure Math. 97, American Mathematical Society, Providence (2018), 597–635. 10.1090/pspum/097.2/01717Search in Google Scholar

Received: 2021-10-15
Revised: 2022-06-01
Published Online: 2022-08-12
Published in Print: 2022-10-01

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